Steiner Minimal Trees: An Introduction, Parallel Computation, and Future Work

Abstract

Minimizing a network’s length is one of the oldest optimization problems in mathematics and, consequently, it has been worked on by many of the leading mathematicians in history. In the mid-seventeenth century a simple problem was posed: Find the point P that minimizes the sum of the distances from P to each of three given points in the plane. Solutions to this problem were derived independently by Fermat, Torricelli, and Cavaliers. They all deduced that either P is inside the triangle formed by the given points and that the angles at P formed by the lines joining P to the three points are all 120°, or P is one of the three vertices and the angle at P formed by the lines joining P to the other two points is greater than or equal to 120°.